Complete Additivity and Modal Incompleteness

In this paper, we tell a story about incompleteness in modal logic. The story weaves together a paper of van Benthem, `Syntactic aspects of modal incompleteness theorems,' and a longstanding open question: whether every normal modal logic can be characterized by a class of completely additive modal algebras, or as we call them, V-BAOs. Using a first-order reformulation of the property of complete additivity, we prove that the modal logic that starred in van Benthem's paper resolves the open question in the negative. In addition, for the case of bimodal logic, we show that there is a naturally occurring logic that is incomplete with respect to V-BAOs, namely the provability logic GLB. We also show that even logics that are unsound with respect to such algebras do not have to be more complex than the classical propositional calculus. On the other hand, we observe that it is undecidable whether a syntactically defined logic is V-complete. After these results, we generalize the Blok Dichotomy to degrees of V-incompleteness. In the end, we return to van Benthem's theme of syntactic aspects of modal incompleteness

Consequences of a Goedel's misjudgment

The fundamental aim of the paper is to correct an harmful way to interpret a Goedel's erroneous remark at the Congress of Koenigsberg in 1930. Despite the Goedel's fault is rather venial, its misreading has produced and continues to produce dangerous fruits, as to apply the incompleteness Theorems to the full second-order Arithmetic and to deduce the semantic incompleteness of its language by these same Theorems. The first three paragraphs are introductory and serve to define the languages inherently semantic and its properties, to discuss the consequences of the expression order used in a language and some question about the semantic completeness: in particular is highlighted the fact that a non-formal theory may be semantically complete despite using a language semantically incomplete. Finally, an alternative interpretation of the Goedel's unfortunate comment is proposed. KEYWORDS: semantic completeness, syntactic incompleteness, categoricity, arithmetic, second-order languages, paradoxesComment: English version, 19 pages. Fixed and improved terminolog

Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference [Book Review]

Can the reassessment of a historical debate contribute to the better understanding of an open philosophical question? The editors of this volume say that it can. The open question concerns the interpretation of quantum mechanics. The historical debate under review is the famous 1927 Solvay conference in Brussels. According to the received view, the standard Copenhagen interpretation was established as the canonical understanding of the new concepts brought about by quantum mechanics during that conference. The conference is remembered, above all, for the famous debate between Bohr and Einstein about the limits and understanding of the quantum uncertainty relations. Again and again, the received view has it, Einstein would come up with ideas for an experiment proving the inconsistency or incompleteness of the new quantum theoretical concepts. And again and again, Bohr would come up with a refutation of Einstein's challenge, proving the Copenhagen interpretation to be consistent and inevitable. But we really know about that debate between Einstein and Bohr only from the latter's own account, published some twenty years later in Paul Arthur Schilpp's Albert Einstein: Philosopher‐Scientist (Open Court, 1949). Contemporary accounts, most importantly a famous letter by Ehrenfest, are less explicit and more equivocal about the debate between Bohr and Einstein

A Method for Reasoning about other Agents\u27 Beliefs from Observations

Traditional work in belief revision deals with the question of what an agent should believe upon receiving new information. We will give an overview about what can be concluded about an agent based on an observation of its belief revision behaviour. The observation contains partial information about the revision inputs received by the agent and its beliefs upon receiving them. We will sketch a method for reasoning about past and future beliefs of the agent and predicting which inputs it accepts and rejects. The focus of this talk will be on different degrees of incompleteness of the observation and variants of the general question we are able to deal with

Fictionalism and the incompleteness problem

This is the final version of the article. It first appeared from Springer via http://dx.doi.org/10.1007/s11229-015-1000-1Modal fictionalists face a problem that arises due to their possible-world story being incomplete in the sense that certain relevant claims are neither true nor false according to it. It has recently been suggested that this incompleteness problem generalises to other brands of fictionalism, such as fictionalism about composite or mathematical objects. In this paper, I argue that these fictionalist positions are particularly threatened by a generalised incompleteness problem since they cannot emulate the modal fictionalists’ most attractive response. I then defend mathematical and compositional fictionalism by showing that the reasons for which the incompleteness problem has been thought to affect them are mistaken. This leads to the question of whether there are other fictionalist positions to which the problem does in fact generalise. I give a general account of the features of a fictionalist position that generate the incompleteness problem and argue that whenever a fictionalist position does exemplify these features then the problem can be addressed in analogy to the modal fictionalists’ preferred response.This research was supported by the Robert Smithson Studentship from the Faculty of Philosophy at the University of Cambridge and by an Analysis Studentship from the Analysis Trust

Gödel, Turing and the Iconic/Performative Axis

1936 was a watershed year for computability. Debates among Gödel, Church and others over the correct analysis of the intuitive concept “human effectively computable”, an analysis at the heart of the Incompleteness Theorems, the Entscheidungsproblem, the question of what a finite computation is, and most urgently—for Gödel—the generality of the Incompleteness Theorems, were definitively set to rest with the appearance, in that year, of the Turing Machine. The question I explore here is, do the mathematical facts exhaust what is to be said about the thinking behind the “confluence of ideas in 1936”? I will argue for a cultural role in Gödel’s, and, by extension, the larger logical community’s absorption of Turing’s 1936 model. As scaffolding I employ a conceptual framework due to the critic Leo Marx of the technological sublime; I also make use of the distinction within the technological sublime due to Caroline Jones, between its iconic and performative modes—a distinction operating within the conceptual art of the 1960s, but serving the history of computability equally well